‘रचना-प्रयोग’ का मुख्य उद्देश्य क्या है?

व्याकरण का व्यावहारिक प्रयोग सिखाना

‘द्वितीयः’ किस प्रकार की संख्या है?

क्रमवाचक (दूसरा)

Write the total surface area and volume of a cylinder of radius r and height h.

Total surface area \(= 2\pi r(r + h)\) and volume \(= \pi r^2 h\).

Write the volume of a cone and of a sphere.

Volume of a cone \(= \dfrac{1}{3}\pi r^2 h\) and volume of a sphere \(= \dfrac{4}{3}\pi r^3\).

Write the curved surface area of a cone with slant height l.

Curved surface area \(= \pi r l\), where \(l = \sqrt{r^2 + h^2}\).

What is meant by a combination of solids? Give an example.

A shape formed by joining two or more basic solids, e.g. a cone mounted on a cylinder (an ice-cream cone) or a hemisphere on a cuboid.

When a solid is melted and recast into another shape, which quantity is conserved?

The volume remains the same, even though the surface area and shape change.

How do you find the surface area of a combined solid?

By adding the areas of the visible (exposed) surfaces of the individual solids that form it.

Find the volume of a hemisphere of radius 3 cm. (Take \(\pi = 22/7\).)

Volume \(= \dfrac{2}{3}\pi r^3 = \dfrac{2}{3} \times \dfrac{22}{7} \times 27 \approx 56.57\) cm\(^3\).

A cone and a cylinder have the same radius and height. What is the ratio of their volumes?

Volume of cone : volume of cylinder \(= \dfrac{1}{3}\pi r^2 h : \pi r^2 h = 1 : 3\).

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Chapter 12: Surface Areas and VolumesClass 10 Mathematics — summary, notes, extra questions & NCERT solutions

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The volume of a sphere of radius r is:

Summary

This chapter extends mensuration to solids and combinations of solids. It revisits the surface areas and volumes of the cuboid, cube, cylinder, cone, sphere and hemisphere, and teaches how to find these quantities for objects formed by joining such solids — for example a cone mounted on a cylinder. Problems also involve converting one solid into another (recasting), where the volume stays constant while the shape changes. These skills apply directly to real objects like containers, tents and toys.

Surface areas of basic solidsVolumes of basic solidsCombination of solidsConversion of solids

Key terms

Surface area

The total area of all the outer faces of a solid.

Volume

The amount of space a solid occupies.

Combination of solids

A shape formed by joining two or more basic solids.

Recasting

Melting and reforming a solid so its volume is conserved.

Extra questions & answers

EXERCISE 12.1

EXERCISE 12.2

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Volume of a cylinder

Radius (cm)Height (cm)

Volume =1539.38cm³

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Surface area

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The total area of all the outer faces of a solid.

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